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Theorem bnj1307 31217
Description: Technical lemma for bnj60 31256. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1307.1 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1307.2 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Assertion
Ref Expression
bnj1307 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Distinct variable groups:   𝑤,𝐵   𝑤,𝑑,𝑥   𝑥,𝑓
Allowed substitution hints:   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑤,𝑓,𝑑)   𝐺(𝑥,𝑤,𝑓,𝑑)   𝑌(𝑥,𝑤,𝑓,𝑑)

Proof of Theorem bnj1307
StepHypRef Expression
1 bnj1307.1 . . 3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
2 bnj1307.2 . . . . . 6 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
32nfcii 2784 . . . . 5 𝑥𝐵
4 nfv 1883 . . . . . 6 𝑥 𝑓 Fn 𝑑
5 nfra1 2970 . . . . . 6 𝑥𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)
64, 5nfan 1868 . . . . 5 𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
73, 6nfrex 3036 . . . 4 𝑥𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
87nfab 2798 . . 3 𝑥{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
91, 8nfcxfr 2791 . 2 𝑥𝐶
109nfcrii 2786 1 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947
This theorem is referenced by:  bnj1311  31218  bnj1373  31224  bnj1498  31255  bnj1525  31263  bnj1523  31265
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